SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Our algorithm for computing the outcome of MIXED BORDA BRANCHING is an extension of an algorithm by Fulkerson [1974] for computing an arbitrary min-cost branchings in a digraph. The algorithm by Fulkerson follows a primal-dual approach and can be divided into two phases, where the first phase characterizes min-cost branchings with the help of a family of subsets of the nodes, and the second phase then constructs one arbitrary min-cost branching. Building upon the first phase, we show that, for every two nodes  $v \in N$  and  $s \in S$ , we can count the number of min-cost branchings that connect the two nodes by applying an extension of the matrix tree theorem [Tutte, 1948, De Leenheer, 2020] and the Markov chain tree theorem [Leighton and Rivest, 1986].
Fulkerson's algorithm. The algorithm (described formally in Algorithm 1) maintains a function  $y:2^{N\cup S}\to \mathbb{N}$  and a subset of the edges  $E_{y}\subseteq E$ . The set  $E_{y}$  captures edges that are tight (w.r.t.  $y$ ), i.e., those  $e\in E$  satisfying
$$
\sum_ {X \subseteq N \cup S: e \in \delta^ {+} (X)} y (X) = c (e).
$$
The algorithm takes as input a delegation graph  $G$  together with a cost function  $c: E \to \mathbb{N}$ .
Algorithm 1 Fulkerson's Algorithm [Fulkerson, 1974, Kamiyama, 2014]
1: Set  $y(X) = 0$  for any set  $X \subseteq N \cup S$  except  $y(\{s\}) = 1$  for any  $s \in S$ ,  $E_y = \emptyset$ 
2: while some node in  $N$  cannot reach any node in  $S$  in the graph  $(N \cup S, E_y)$  do
3: let  $X \subseteq N$  be a strongly connected component in  $(N \cup S, E_y)$  with  $\delta^+(X) \cap E_y = \emptyset$ 
4: set  $y(X)$  to minimum value such that some edge in  $\delta^+(X)$  is tight, add tight edges to  $E_y$ 
5:  $\mathcal{F} = \{X \subseteq N \cup S \mid y(X) > 0\}$ 
6: return  $(\mathcal{F}, E_y, y)$
Since  $c(e) \geq 1$  for all  $e \in E$ , the output  $\mathcal{F}$  contains all singleton sets induced by nodes in  $N \cup S$  and beyond that subsets of  $N$ . In the following we show several structural insights that are crucial for the construction of our algorithm. While statements (ii) and (iii) have been proven in similar forms by Fulkerson [1974], we prove all of Lemma 2 in Appendix A for completeness.
Lemma 2 ( $\star$ ). Let  $(G, c)$  be a delegation graph and let  $(\mathcal{F}, E_y, y)$  be the output of Algorithm 1. Then:
(i) For every  $(G, c)$ , the output of the algorithm is unique, i.e., it does not depend on the choice of the strongly connected component in line 3.
(ii)  $\mathcal{F}$  is laminar, i.e., for any  $X,Y\in \mathcal{F}$  it holds that either  $X\subseteq Y$ $Y\subseteq X$  , or  $X\cap Y = \emptyset$
(iii) Branching  $B$  in  $(G,c)$  is min-cost iff (a)  $B\subseteq E_y$ , and (b)  $|B\cap \delta^{+}(X)| = 1$  for all  $X\in \mathcal{F},X\subseteq N$
(iv) For every  $X \in \mathcal{F}$ , an in-tree  $T$  in  $G[X] = (X, E[X])$ , where  $E[X] = \{(u, v) \in E \mid u, v \in X\}$ , is min-cost iff  $(a) T \subseteq E_{y}$ , and  $(b) |T \cap \delta^{+}(Y)| = 1$  for all  $Y \in \mathcal{F}$  such that  $Y \subset X$ .
Intuition and notation for Algorithm 2. For our algorithm, the crucial statements in Lemma 2 are (ii) and (iii). First, because  $\mathcal{F}$  forms a laminar family, there exists a natural tree-like structure among the sets. We say that a set  $Y\in \mathcal{F}$  is a child of a set  $X\in \mathcal{F}$ , if  $Y\subset X$  and there does not exist a  $Z\in \mathcal{F}$  such that  $Y\subset Z\subset X$ . Moreover, for some  $X\in \mathcal{F}$  or  $X = N\cup S$ , we define  $G_{X} = (V_{X},E_{X})$  as the tight subgraph that is restricted to the node set  $X$  and contracts all children of  $X$ . Formally,  $V_{X} = \{Y\mid Y$  is a child of  $X\}$  and  $E_{X} = \{(Y,Y^{\prime})\mid (u,v)\in E_{y},u\in Y,v\in Y^{\prime}$ , and  $Y,Y^{\prime}\in V_{X}\}$ . In the following we first focus on the graph corresponding to the uppermost layer of the hierarchy, i.e.,  $G_{X}$  for  $X = N\cup S$ . Now, statement (iii) implies that every min-cost branching  $B$  in  $G$  leaves every child of  $X$  exactly once and only uses tight edges. Hence, if we project  $B$  to an edge set  $\hat{B}$  in the contracted graph  $G_{X}$ , then  $\hat{B}$  forms a branching in  $G_{X}$ . However, there may exist many min-cost branchings in  $\hat{G}$  that map to the same branching in  $G_{X}$ . The crucial insight is that we can construct a weight function  $w_{X}$  on the edges of  $G_{X}$ , such that the weighted number of branchings in  $G_{X}$  equals the number of min-cost branchings in  $G$ . This function is constructed by calculating for every child
<sup>9</sup>We slightly adjust the algorithm, as Fulkerson [1974] studies directed out-trees and assumes one sink only.
$$
w _ {X _ {1}} \left(\mathcal {T} _ {Y _ {1}} \left(G _ {X _ {1}}\right)\right) = 1
$$
$$
w _ {X _ {1}} \left(\mathcal {T} _ {Y _ {2}} \left(G _ {X _ {1}}\right)\right) = 3
$$
$Y$  of  $X$  and every node  $v \in Y$ , the number of min-cost  $v$ -trees inside the graph  $G[Y] = (Y, E[Y])$ , where  $E[Y] = \{(u, v) \in E \mid u, v \in Y\}$ . This number, denoted by  $t_Y(v)$ , can be computed by recursively applying the matrix tree theorem. Coming back to the graph  $G_X$ , however, we need a more powerful tool since we need to compute the (relative) number of weighted branchings in  $G_X$  connecting any node to a sink node. Thus, we introduce the Markov chain tree theorem (in a slightly modified version so that it can deal with Markov chains induced by weighted digraphs).
For a weighted digraph  $(G, w)$  we define the corresponding Markov chain  $(G', P)$  as follows: The digraph  $G'$  is derived from  $G$  by adding self-loops, and for  $(u, v) \in E(G')$  we let
$$
P _ {u, v} = \left\{ \begin{array}{l l} \frac {w (u , v)}{\Delta} & \text {i f} u \neq v \\ 1 - \frac {\sum_ {e \in \delta^ {+} (u)} w (e)}{\Delta} & \text {i f} u = v, \end{array} \right.
$$
where  $\Delta = \max_{v\in V}\sum_{e\in \delta^{+}(v)}w(e)$
Lemma 3 (Markov chain tree theorem (Leighton and Rivest [1986])) Consider a weighted digraph  $(G, w)$  and the corresponding Markov chain  $(G', P)$  and let  $Q = \lim_{\tau \to \infty} \frac{1}{\tau} \sum_{i=0}^{\tau} P^{\tau}$ . Then, the entries of the matrix  $Q$  are given by
$$
Q _ {u, v} = \frac {\sum_ {B \in \mathcal {B} _ {u , v}} \prod_ {e \in B} P _ {e}}{\sum_ {B \in \mathcal {B}} \prod_ {e \in B} P _ {e}} = \frac {\sum_ {B \in \mathcal {B} _ {u , v}} \prod_ {e \in B} w (e)}{\sum_ {B \in \mathcal {B}} \prod_ {e \in B} w (e)} = \frac {\sum_ {B \in \mathcal {B} _ {u , v}} w (B)}{\sum_ {B \in \mathcal {B}} w (B)}.
$$
We formalize Algorithm 2, which takes as input a delegation graph  $(G,c)$  and, in contrast to the intuition above, works in a bottom-up fashion. We refer to Figure 2 for an illustration.
Theorem 4 ( $\star$ ). Algorithm 2 returns MIXED BORDA BRANCHING and runs in poly(n).
Proof sketch. The main part of the proof, shows by induction over the laminar hierarchy of  $\mathcal{F}$ , that for every  $X \in \mathcal{F}$  and  $v \in X$ , the value  $t_X(v)$  equals the number of min-cost  $v$ -trees in  $G[X]$ . Given that this is true, one can show that  $w_X(\mathcal{B}_{Y,\{s\}}(G_X))$  equals the number of min-cost branchings in  $G$  that connect any node in  $v \in Y$  to the sink  $s$  and that  $w_X(\mathcal{B}(G_X))$  equals the number of all min-cost branchings in  $G$ . Hence, we can utilize the Markov chain tree theorem on the graph  $G_X$  to compute
Algorithm 2 Computation of MIXED BORDA BRANCHING
1: compute  $(\mathcal{F},E_y,y)$  , set  $\mathcal{F}' = \mathcal{F}\cup \{N\cup S\}$  and label its elements "unprocessed"  $\triangleright$  Algorithm 1   
2:  $t_{\{v\}}(v)\gets 1$  for all  $v\in N\cup S$  , label singletons as "processed"   
3: do pick unprocessed  $X\in \mathcal{F}'$  for which all children are processed, label  $X$  as "processed"   
4: set  $w_{X}(Y,Y^{\prime})\leftarrow \sum_{(u,v)\in E_{y}\cap (Y\times Y^{\prime})}t_{Y}(u)$  , for all children  $Y$  and  $Y^\prime$  of  $X$    
5: if  $X\neq N\cup S$  then   
6: for all children  $Y$  of  $X$  do   
7:  $t_X(v)\gets w_X(\mathcal{T}_Y(G_X))\cdot t_Y(v)$  for all  $v\in Y$ $\triangleright$  Lemma 1   
8: else compute absorbing probability matrix  $Q$  for the Markov chain  $\triangleright$  Lemma 3 corresponding to  $(G_X,w_X)$    
9: while  $X\neq N\cup S$    
10: return for all  $v\in N$  and  $s\in S$  ..  $A_{v,s}\gets Q_{Y_v,\{s\}}$  , where  $Y_{v}$  child of  $N\cup S$  with  $v\in Y_v$
the outcome of MIXED BORDA BRANCHING.
As for the running time, the main observation is that the computation of the number of branchings in a weighted digraph can be done in time logarithmic in the highest weight of an edge (and polynomial in the number of edges). Since all of our weights are bounded by the maximum number of branchings in the original graph (which is bounded by  $|N|^{|N|}$ ), we can show the polynomial running time.
6 Equivalence of MIXED BORDA BRANCHING and RANDOM WALK RULE
With the help of the Markov chain tree theorem, as stated in Section 5, we can show the equivalence of the RANDOM WALK RULE and MIXED BORDA BRANCHING.
Theorem 5. Let  $(G, c)$  be a delegation graph and  $A$  and  $\hat{A}$  be the assignments returned by MIXED BORDA BRANCHING and the RANDOM WALK RULE, respectively. Then,  $A = \hat{A}$ .
Proof. Let  $v \in N$  and  $s \in S$ , then,
$$
\begin{array}{l} \hat {A} _ {v, s} = \lim  _ {\varepsilon \rightarrow 0} \left(\frac {1}{\tau} \sum_ {\tau = 1} ^ {\infty} (P ^ {(\varepsilon)}) ^ {\tau}\right) _ {v, s} = \lim  _ {\varepsilon \rightarrow 0} \frac {\sum_ {B \in \mathcal {B} _ {v , s}} \prod_ {e \in B} P _ {e} ^ {(\varepsilon)}}{\sum_ {B \in \mathcal {B}} \prod_ {e \in B} P _ {e} ^ {(\varepsilon)}} = \lim  _ {\varepsilon \rightarrow 0} \frac {\sum_ {B \in \mathcal {B} _ {v , s}} \prod_ {(u , v) \in B} \frac {\varepsilon^ {c (u , v)}}{\bar {\varepsilon} _ {u}}}{\sum_ {B \in \mathcal {B}} \prod_ {(u , v) \in B} \frac {\varepsilon^ {c (u , v)}}{\bar {\varepsilon} _ {u}}} \\ = \lim  _ {\varepsilon \rightarrow 0} \frac {\prod_ {u \in N} (\bar {\varepsilon} _ {u}) ^ {- 1} \sum_ {B \in \mathcal {B} _ {v , s}} \varepsilon^ {c (B)}}{\prod_ {u \in N} (\bar {\varepsilon} _ {u}) ^ {- 1} \sum_ {B \in \mathcal {B}} \varepsilon^ {c (B)}} = \frac {\sum_ {B \in \mathcal {B} _ {v , s} ^ {*}} 1}{\sum_ {B \in \mathcal {B} ^ {*}} 1} = A _ {v, s}. \\ \end{array}
$$
We first use the definition of the RANDOM WALK RULE, and then apply the Markov chain tree theorem (Lemma 3) for fixed  $\varepsilon \in (0,1]$  to obtain the second equality. For the third equality, we use the definition of  $P^{(\varepsilon)}$ , and then factor out the normalization factor  $\bar{\varepsilon}_u$  for every  $u \in N$ . For doing so, it is important to note that for every  $v \in N$  and  $s \in S$ , every branching in  $\mathcal{B}_{v,s}$  and every branching in  $\mathcal{B}$  contains exactly one outgoing edge per node in  $N$ . We also remind the reader that  $c(B) = \sum_{e \in B} c(e)$ . Finally, we resolve the limit in the fifth equality by noting that the dominant parts of the polynomials are those corresponding to min-cost (BORDA) branchings.
Given Theorem 5, we can interpret Algorithm 2 as an algorithm computing limit absorbing probabilities of a class of parametric Markov chains. We explain this reinterpretation in Appendix B.
7 Axiomatic Analysis
In this section, we generalize and formalize the axioms mentioned in Section 1 and show that the RANDOM WALK RULE (and hence MIXED BORDA BRANCHING) satisfies all of them. In particular, our version of confluence (copy-robustness, respectively) reduces to (is stronger than, respectively) the corresponding axiom for the non-fractional case by Brill et al. [2022] (see Appendix C.1). We first define anonymity, which prescribes that a delegation rule should not make decisions based on the identity of the voters. Given a digraph with a cost function  $(G,c)$  and a bijection  $\sigma :V(G)\to V(G)$ , we define the graph  $\sigma ((G,c))$  as  $(G^{\prime},c^{\prime})$ , where  $V(G^{\prime}) = V(G)$ ,  $E(G^{\prime}) = \{(\sigma (u),\sigma (v)\mid (u,v)\in E(G)\}$  and  $c^\prime (\sigma (u),\sigma (v)) = c(u,v)$  for each edge  $(u,v)\in E(G)$ .
Anonymity: For any delegation graph  $(G,c)$ , any bijection  $\sigma :V(G)\to V(G)$ , and any  $v\in N,s\in S$ , it holds that  $A_{v,s} = A_{\sigma (v),\sigma (s)}^{\prime}$ , where  $A$  and  $A^{\prime}$  are the outputs of the delegation rule for  $(G,c)$  and  $\sigma ((G,c))$ , respectively.
Theorem 6 ( $\star$ ). The RANDOM WALK RULE satisfies anonymity.
We now define copy-robustness, which intuitively demands that if a delegating voter  $v \in N$  decides to cast their vote instead, the total voting weight of  $v$  and all its representatives should not change. This lowers the threat of manipulation by a voter deciding whether to cast or delegate their vote depending on which gives them and their representatives more total voting weight. This axiom was introduced (under a different name) by Behrens and Swierczek [2015] and defined for non-fractional delegation rules in [Brill et al., 2022]. We strengthen $^{10}$  and generalize the version of Brill et al. [2022].
Copy-robustness: For every delegation graph  $(G, c)$  and delegating voter  $v \in N$ , the following holds: Let  $(\hat{G}, c)$  be the graph derived from  $(G, c)$  by removing all outgoing edges of  $v$ , let  $A$  and  $\hat{A}$  be the output of the delegation rule for  $(G, c)$  and  $(\hat{G}, c)$ , respectively and let  $S_v = \{s \in S \mid A_{v,s} > 0\}$  be the set of representatives of  $v$  in  $(G, c)$ . Then  $\sum_{s \in S_v} \pi_s(A) = \pi_v(\hat{A}) + \sum_{s \in S_v} \pi_s(\hat{A})$ .
Theorem 7  $(\star)$ . The RANDOM WALK RULE satisfies copy-robustness.
Proof sketch. We show a statement that is slightly stronger than the condition for copy-robustness. Namely, the voting weight of every casting voter from  $S \setminus S_v$  remains equal when  $v$  changes from being a delegating voter to becoming a casting voter. Let  $(G_X, w_X)$ , with  $X = N \cup S$ , be the contracted graph constructed in Algorithm 2 for  $(G, c)$ . Analogously, let  $(\hat{G}_X, \hat{w}_X)$  be the contracted graph constructed for  $(\hat{G}, c)$ . In order to argue about their relation, we first show: If  $y$  and  $\hat{y}$  are the functions returned by Algorithm 1 for  $G$  and  $\hat{G}$ , respectively, then  $\hat{y}(Y) = y(Y)$  for every set  $Y \subseteq 2^{N \setminus \{v\}}$  not containing  $v$ ,  $\hat{y}(\{v\}) = 1$ , and  $\hat{y}(Y) = 0$  for all other sets containing  $v$ . Now, let  $Y_v$  be the node in  $G_X$  containing  $v$  and let  $\mathcal{U} \subseteq V(G_X)$  be the subset of nodes in  $G_X$  that are not reachable from  $Y_v$ . We argue that for any  $s \in S \setminus S_v$ , the Markov chain induced by  $(G_X, w_X)$  can only reach  $\{s\}$  from a starting node in  $\mathcal{U}$ . However, for all nodes in  $\mathcal{U}$ , all of their walks to any  $\{s\}$ ,  $s \in S \setminus S_v$  are still existent in the graph  $(\hat{G}_X, \hat{w}_X)$ , and still have the same weight. Hence, the voting weight of any  $s \in S \setminus S_v$  remains unchanged when moving from  $G$  to  $\hat{G}$ .
To capture the requirement that the voting weight of different voters is assigned to casting voters in a "consistent" way, Brill et al. [2022] define confluence as follows: A delegation rule selects, for every voter  $u \in N$ , one walk in the delegation graph starting in  $u$  and ending in some sink  $s \in S$ , and assigns voter  $u$  to casting voter  $s$ . A delegation rule satisfies confluence, if, as soon as the walk of  $u$  meets some other voter  $v$ , the remaining subwalk of  $u$  equals the walk of  $v$ .<sup>11</sup> Below, we provide a natural generalization of the property by allowing a delegation rule to specify a probability distribution over walks. Then, conditioned on the fact that the realized walk of some voter  $u$  meets voter  $v$ , the probability that  $u$  reaches some sink  $s \in S$  should equal the probability that  $v$  reaches  $s$ .
Confluence: For every delegation graph  $(G,c)$ , there exists a probability distribution  $f_{v}$  for all  $v\in N$  over the set of walks in  $G$  that start in  $v$  and end in some sink, which is consistent with the assignment  $A$  of the delegation rule (i.e.,  $\mathbb{P}_{W\sim f_v}[s\in W] = A_{v,s}$  for all  $v\in N,s\in S$ ), and,
$$
\mathbb {P} _ {W \sim f _ {u}} [ s \in W \mid v \in W ] = \mathbb {P} _ {W \sim f _ {v}} [ s \in W ] \quad \text {f o r a l l} u, v \in N, s \in S.
$$
Note that the requirement that  $A_{v,s} = \mathbb{P}_{W\sim f_v}[s\in W]$  implies that for any  $v\in V$  and  $s\in S$  we can have  $A_{v,s} > 0$  only if there is a path from  $v$  to  $s$  in  $G$ .
Theorem 8 ( $\star$ ). The RANDOM WALK RULE satisfies confluence.
Proof sketch. One can verify that every delegation rule that can be formalized via a Markov chain on the delegation graph  $(G, c)$  satisfies confluence. In Section 5, we showed that the RANDOM WALK RULE can be computed by solving a Markov chain  $(G_X, P)$  on the contracted graph  $G_X$  (for  $X = N \cup S$ ). We utilize  $(G_X, P)$  to define a probability distribution over walks in  $G$  that satisfies confluence: Every walk in  $G$  can be mapped to a walk in  $G_X$  (by ignoring edges inside children of  $X$ ), but there may exist many walks in  $G$  that map to the same walk in  $G_X$ . We pick, for every walk
<sup>10</sup>Brill et al. [2022] restrict the condition to voters that have a direct connection to their representative.
<sup>11</sup>Brill et al. [2022] assume paths instead of walks. The two definitions are equivalent (see Appendix C.1).
Figure 3: Situation in the proof of Theorem 9. Solid edges correspond to first-choice delegations, dashed edges to second-choice delegations.
$\hat{W}$  in  $G_{X}$ , one representative walk  $W$  in  $G$  and give it the same probability as  $\hat{W}$  in  $(G_{X},P)$ . All other walks in  $G$  get zero probability. When picking the representative walks, we can ensure that for every two nodes  $u,v\in N\cup S$ , the probability that the walk of  $u$  reaches  $v$  equals the probability that  $Y_{u}$  reaches  $Y_{v}$  in  $(G_{X},P)$ , where  $Y_{u}$  and  $Y_{v}$  are the nodes in  $V(G_{X})$  containing  $u$  and  $v$ , respectively. For the constructed probability distribution, we then show that the confluence condition is met.
With the formal definition of confluence, anonymity, and copy-robustness we can now show that these properties altogether are impossible to achieve in the non-fractional case. Recall, that a non-fractional delegation rule is defined as a delegation rule, that returns assignments  $A \in \{0,1\}^{N \times S}$ .
Theorem 9. No non-fractional delegation rule satisfies confluence, anonymity, and copy-robustness.
Proof. Consider the graph  $(G_1, c_1)$  in Figure 3. There are four non-fractional assignments in  $(G_1, c_1)$ : Both  $v_1$  and  $v_2$  can either be assigned  $v_3$  or  $v_4$ . Suppose a rule chooses assignment  $A$  with  $A_{v_1, v_4} = A_{v_2, v_3} = 1$ . This rule cannot satisfy confluence, as any walk from  $v_2$  to  $v_3$  includes  $v_1$  and confluence requires  $1 = A_{v_2, v_3} = \mathbb{P}_{W \sim f_{v_2}}[v_3 \in W] = \mathbb{P}_{W \sim f_{v_2}}[v_3 \in W \mid v_1 \in W] = \mathbb{P}_{W \sim f_{v_1}}[v_3 \in W] = A_{v_1, v_3} = 0$ . Now, suppose a delegation rule chooses assignment  $A$  with  $A_{v_1, v_3} = A_{v_2, v_3} = 1$ . We define the bijection  $\sigma$  mapping  $v_1$  to  $v_2$ ,  $v_2$  to  $v_1$ ,  $v_3$  to  $v_4$  and  $v_4$  to  $v_3$ . Then,  $\sigma((G_1, c)) = (G_1, c)$  and thus  $A_{v_1, v_3}' = A_{v_2, v_3}' = 1$  in the assignment  $A'$  that the rule chooses for  $\sigma((G_1, c))$ . This contradicts anonymity, since  $1 = A_{v_1, v_3} \neq A_{\sigma(v_1), \sigma(v_3)}' = A_{v_2, v_4}' = 0$ . We can make the same argument in the case of  $A_{v_1, v_4} = A_{v_2, v_4} = 1$ . For any rule satisfying anonymity and confluence the chosen assignment  $A$  must therefore have  $A_{v_1, v_3} = A_{v_2, v_4} = 1$ .
The above arguments are independent of the cost function  $c$ , so long as we have  $c(v_{1}, v_{2}) = c(v_{2}, v_{1})$  and  $c(v_{1}, v_{3}) = c(v_{2}, v_{4})$ , needed for the equality of  $\sigma((G_{1}, c))$  and  $(G_{1}, c)$ . Thus, any rule satisfying anonymity and confluence must choose the assignment  $A$  with  $A_{v_{1}, v_{2}} = A_{v_{3}, v_{4}} = 1$  for  $(G_{2}, c_{2})$ . We modify  $(G_{1}, c_{1})$  by making  $v_{2}$  a casting voter (as in the definition of copy-robustness) and retrieve  $(G_{3}, c_{3})$ . Copy robustness requires that the assignment from  $v_{1}$  to  $v_{4}$  in  $G_{1}$  (which is zero) must be the same as the sum of assignments from  $v_{1}$  to  $v_{4}$  and  $v_{2}$ . Thus, we have  $A_{v_{1}, v_{3}} = 1$  for the assignment  $A$ , that any confluent, anonymous, and copy-robust rule chooses for  $(G_{3}, c_{3})$ . However, we can also construct  $(G_{3}, c_{3})$  from  $(G_{2}, c_{2})$  by making  $v_{3}$  a casting voter. Then, analogously, copy-robustness requires  $A_{v_{1}, v_{2}} = 1$  for the assignment of  $(G_{3}, c_{3})$ , leading to a contradiction.
Since the RANDOM WALK RULE (and thus MIXED BORDA BRANCHING) satisfies generalizations of the three axioms, the above impossibility is due to its restriction to non-fractional rules.
8 Concluding Remarks
We generalized the setting of liquid democracy with ranked delegations to allow for fractional delegation rules. Beyond that, we presented a delegation rule that can be computed in polynomial time and satisfies a number of desirable properties. A natural follow-up question is to understand the entire space of delegation rules satisfying these properties.
Fractional delegations have been recently implemented (see electric vote) and studied by Degrave [2014] and Bersetche [2022]. In contrast to our setting, these approaches let agents declare a desired distribution over their delegates (instead of rankings). We remark that one could easily combine the two approaches by letting agents declare their desired split within each equivalence class of their ranking. Our algorithm can be extended for this setting (see Appendix B).
There exists a line of research which aims to understand liquid democracy from an epistemic viewpoint [Kahng et al., 2021, Caragiannis and Micha, 2019, Halpern et al., 2023]. Here, many of the negative results stem from the fact that voting weight is concentrated on few casting voters. Since, intuitively, ranked delegations can help to distribute the voting weight more evenly, it would be interesting to study these through the epistemic lens.
This work was supported by the Deutsche Forschungsgemeinschaft (under grant BR 4744/2-1), the Centro de Modelamento Matemático (CMM) (under grant FB210005, BASAL funds for center of excellence from ANID-Chile), ANID-Chile (grant ACT210005), and the Dutch Research Council (NWO) (project number 639.023.811, VICI "Collective Information"). Moreover, this work was supported by the National Science Foundation under Grant No. DMS-1928930 and by the Alfred P. Sloan Foundation under grant G-2021-16778, while Ulrike Schmidt-Kraepelin was in residence at the Simons Laufer Mathematical Sciences Institute (formerly MSRI) in Berkeley, California, during the Fall 2023 semester.
We would like to thank Markus Brill for suggesting the setting to us as well as insightful discussions. Moreover, we thank Jannik Matuschke for helpful discussions on min-cost branchings. Also, we thank our colleagues from Universidad de Chile, Martin Lackner, and Théo Delemazure for their valuable feedback.